Brick texture with any offset other than 1/2?
The trick to achieving different brick offsets with the standard Brick texture is to additionally offset each individual row - instead of allowing the brick texture to provide the offset. This can be achieved as follows :

Note that the Brick Texture is set with an Offset set to zero - this results in no offset for each subsequent layer of bricks. Additionally, the Scale and Brick Width and Row Height are being set by Value nodes - so we can use those values in our calculations (allowing the texture to still be adjusted without breaking our row offsets).
The key to the calculation is to use the Y coordinate to determine the 'row' of bricks. This is achieved using the Snap node (this will increment by 1 for each full row of bricks). The offset is then essentially calculated based on the row number (from the Snap) multiplied by 1/'Offset Fraction' - but with some additional maths to take account of the Brick Width and Scale.
With Offset Fraction set to '3' we get a repeating pattern every 3 rows - so offset by 1/3rd of a brick :